Lipschitz Regularity for Censored Subdiffusive Integro-Differential Equations with Superfractional Gradient Terms

TL;DR

This paper establishes Lipschitz regularity for censored nonlocal integro-differential equations with superfractional gradient terms, enabling well-posedness and long-term analysis of related elliptic and parabolic problems.

Contribution

It provides the first Lipschitz regularity results for censored nonlocal equations with superfractional gradient terms, and applies these to prove comparison principles and analyze long-term behavior.

Findings

Lipschitz regularity for stationary solutions

Comparison principles for well-posedness

Analysis of steady and ergodic long-term behavior

Abstract

In this paper we are interested in integro-differential elliptic and parabolic equations involving nonlocal operators with order less than one, and a gradient term whose coercivity growth makes it the leading term in the equation. We obtain Lipschitz regularity results for the associated stationary Dirichlet problem in the case when the nonlocality of the operator is confined to the domain, feature which is known in the literature as censored nonlocality. As an application of this result, we obtain strong comparison principles which allow us to prove the well-posedness of both the stationary and evolution problems, and steady/ergodic large time behavior for the associated evolution problem.